The tension process of single crystal Cu nano–rods with different cross section shapes were simulated by molecular dynamics at atomic scale. Based on centrosymmetry parameter method and combined with the dislocation nucleation theory, the effect of cross–section shape, cross–sectional area and slenderness ratio on the tensile mechanical properties of the nano–rods were analyzed, and the scale dependency of tensile mechanical properties of the single crystal Cu nano–rods has been revealed. The results show that after first yield, the nano–rods produce plastic deformation under the "dislocation nucleation–extended dislocation and sliding–lattice atom cross–slip" mechanism of the alternating cycle. The geometry of cross-section has negligible effects on the tensile initial plasticity of the nano–rods, while it shows apparent effects on the tensile mechanical properties. With the increase of cross–sectional area, two types of nano–rods have the phenomenon of early yield point, yield strength decreases and young’modulus increases. Compared with that of the square cross–sectional nano–rod, the variable rate of yield stess of the circular cross-sectional nano–rod is smaller while the variable rate of young’s modulus is lager. As the cross–sectional area increases to 500 nm2, the young’s modulus of the two types of nano–rods become stable, and is close to the theoretical value of 84 GPa. Moreover, the slenderness ratio of the nano–rods has a slight effect on the tensile mechanical properties when the simulation size increased.